🤖 AI Summary
This work proposes an efficient numerical method for computing high-order Lie derivatives, which are essential in nonlinear system analysis yet computationally expensive when evaluated symbolically at increasing orders. By establishing a factorial-scaled equivalence between high-order Lie derivatives and the Taylor coefficients—computed along system trajectories and incorporating variational matrices—the approach circumvents symbolic manipulation entirely. The implementation leverages MATLAB’s ADTAYL package to automate differentiation and variational matrix computation, yielding substantial gains in computational efficiency. Experimental results on a gantry crane model demonstrate speedups of several orders of magnitude compared to MATLAB’s Symbolic Math Toolbox, highlighting the method’s practical advantage for high-order analyses in nonlinear control and dynamics.
📝 Abstract
High-order Lie derivatives are essential in nonlinear systems analysis. If done symbolically, their evaluation becomes increasingly expensive as the order increases. We present a compact and efficient numerical approach for computing Lie derivatives of scalar, vector, and covector fields using the MATLAB ADTAYL package. The method exploits a fact noted by R\"obenack: that these derivatives coincide, up to factorial scaling, with the Taylor coefficients of expressions built from a Taylor expansion about a trajectory point and, when required, the associated variational matrix. Computational results for a gantry crane model demonstrate orders of magnitude speedups over symbolic evaluation using the MATLAB Symbolic Math Toolbox.